POTE tuning: Difference between revisions
→Computer Program for TE and POTE: integrated call into program listing otherwise it would be hard to input |
m Avoid bolded links; various formatting improvements |
||
| Line 1: | Line 1: | ||
'''POTE tuning''' is the short form of ''' | '''POTE tuning''' is the short form of '''pure-octaves Tenney-Euclidean tuning''', a good choice for a standard tuning enforcing a just 2/1 octave. This page focuses on the computation. For more mathematical backgrounds, see [[Tenney-Euclidean tuning]]. | ||
== Computing TE and POTE tuning == | == Computing TE and POTE tuning == | ||
The TE and POTE tuning for a [[ | The TE and POTE tuning for a [[mapping|map matrix]] such as M = [{{val|1 0 2 -1}}, {{val|0 5 1 12}}] (the map for 7-limit [[Magic family|magic]], which consists of a linearly independent list of [[val]]s defining magic) can be found as follows: | ||
# Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log<sub>2</sub>''p'' on the diagonal; in other words the diagonal is [1 1/log<sub>2</sub>3 1/log<sub>2</sub>5 1/log<sub>2</sub>7]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [{{val|1 0 2/ | # Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log<sub>2</sub>''p'' on the diagonal; in other words the diagonal is [1 1/log<sub>2</sub>3 1/log<sub>2</sub>5 1/log<sub>2</sub>7]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [{{val| 1 0 2/log<sub>2</sub>5 -1/log<sub>2</sub>7 }}, {{val| 5/log<sub>2</sub>3 1/log<sub>2</sub>5 12/log<sub>2</sub>7 }}] | ||
# Find the pseudoinverse of the matrix V<sup>+</sup> = V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>. | # Find the pseudoinverse of the matrix V<sup>+</sup> = V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>. | ||
# Find the TE generators g = {{val|1 1 1 1}}V<sup>+</sup>. | # Find the TE generators g = {{val| 1 1 1 1 }}V<sup>+</sup>. | ||
# Find the TE tuning map: T = gV. | # Find the TE tuning map: T = gV. | ||
# Find the POTE generators g<nowiki/>' = g/T<sub>1</sub>; in other words g scalar divided by the first entry of T. | # Find the POTE generators g<nowiki/>' = g/T<sub>1</sub>; in other words g scalar divided by the first entry of T. | ||
| Line 13: | Line 13: | ||
If you carry out these operations, you should find | If you carry out these operations, you should find | ||
* V ~ [{{val|1 0 0.861 -0.356}}, {{val|0 3.155 0.431 4.274}}] | * V ~ [{{val| 1 0 0.861 -0.356 }}, {{val| 0 3.155 0.431 4.274 }}] | ||
* g ~ {{val|1.000902 0.317246}} | * g ~ {{val| 1.000902 0.317246 }} | ||
* g<nowiki/>' ~ {{val|1.000000 0.316960}} | * g<nowiki/>' ~ {{val| 1.000000 0.316960 }} | ||
The tuning of the POTE [[generator]] corresponding to the mapping M is therefore 0.31696 octaves, or 380.352 cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, but the POTE tuning can still be found in this way for mappings defining higher rank temperaments. The method can be generalized to subgroup temperaments so long as the group contains 2 by [[ | The tuning of the POTE [[generator]] corresponding to the mapping M is therefore 0.31696 octaves, or 380.352 cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, but the POTE tuning can still be found in this way for mappings defining higher rank temperaments. The method can be generalized to subgroup temperaments so long as the group contains 2 by [[Lp tuning|POL2 tuning]]. | ||
=== Computer Program for TE and POTE === | === Computer Program for TE and POTE === | ||
| Line 61: | Line 61: | ||
</pre> | </pre> | ||
[[Category: | [[Category:Glossary]] | ||
[[Category: | [[Category:Practical help]] | ||
[[Category: | [[Category:Tuning]] | ||
[[Category: | [[Category:Tuning technique]] | ||